COMPLETE NORMALITY AND COUNTABLE COMPACTNESS

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1 Topology Proceedings Vol 17, 1992 COMPLETE NORMALITY AND COUNTABLE COMPACTNESS PETER J. NYIKOS, BORIS SHAPIROVSKIĬ, ZOLTÁN SZENTMIKLÓSSY AND BOBAN VELIČKOVIĆ One of the classical separation axioms of topology is complete normality. A topological space X is completely normal if for every pair of subsets A and B of X which are separated (i.e. Ā B = = A B) there are disjoint open sets containing A and B respectively. A standard exercise is to show that this is equivalent to hereditary normality. We will refer to completely normal Hausdorff spaces as T 5 spaces. Until now it has been somewhat of a mystery how wellbehaved countably compact T 5 spaces can be. Under Gödel s axiom of constructibility (V = L) they can be quite pathological: in [1] there is a V = L construction of a compact T 5 space X of cardinality 2 c in which every subspace is separable, yet the space has no infinite closed 0-dimensional subspaces; in particular, X has no nontrivial convergent sequences. Our main results show that these spaces are much better behaved under the Proper Forcing Axiom (PFA), introduced in [2]. It implies that every countably compact T 5 space is sequentially compact. [The consistency of this is new even for the compact case.] Much more strongly, it implies that every countable subset of a countably compact T 5 space has compact, Fréchet-Urysohn closure. [A space is called Fréchet-Urysohn if whenever a point x is in the closure of a subset A, then there is a sequence from A converging to x.] Hence, in particular, a separable subspace can have cardinality at most c. 395

2 396 NYIKOS, SHAPIROVSKIĬ, VELIČKOVIĆ A striking corollary of this and the Tychonoff theorem is that, under PFA, the product of any number of countably compact T 5 spaces is countably compact, although the T 5 property may be lost. This establishes the set-theoretic independence of the productivity of countable compactness when the factors are T 5 ; earlier, J. Vaughan [3] showed the consistency of the existence of a family of sequentially compact perfectly normal (T 6 ) spaces whose product is not countably compact. We have also shown that the PFA implies every compact T 5 space is pseudo-radial of chain-net order 2. [A space is called pseudo-radial if closures can be obtained by iterating the operation of taking limits of well-ordered nets.] In fact, the closure of any subset A can be taken by taking the set Â of all limits of convergent sequences in A, and adding to Â the set of all points which are the limit of a well-ordered net in Â. The proofs of these results use the following reduction theorem. For the definitions of countably tight and free sequence see [4], [5], or [6]. Separable is synonymous with having a countable dense subset. Theorem 1 [4] The following are equivalent. (a) Every separable, T 5, compact space is countably tight. (b) Every free sequence in a separable, T 5, countably compact space is countable. (c) A separable, T 5, countably compact space cannot contain ω 1. (d) No version of γn is T 5. Here γn is the generic symbol for a locally compact Hausdorff space X with a countable dense set of isolated points, identified with the set N of positive integers, such that X\N is homeomorphic to ω 1. We will also identify X\N with ω 1 using a definition of N that makes it disjoint from ω 1. Baumgartner and the first author have independently observed that using only the usual axioms of set theory one can construct

3 T 5 AND COUNTABLE COMPACTNESS 397 versions of γn in which the union of N with the successor ordinals is not normal. On the other hand, it is also consistent that there are other versions which are T 5 ([4, 12]). One of our main results is that this cannot happen under the following version of the Open Coloring Axiom (OCA): If X is a separable metric space and [X] 2 = K 0 K 1 is a partition with K 0 open in the product topology then either there exists an uncountable 0-homogeneous subset of X, of else X can be covered by countably many 1-homogeneous sets. As usual, [A] 2 stands for the collection of two-element subsets of A. A subset H of X is called i-homogeneous if [H] 2 K i. In saying K 0 is open in the product topology, what we really mean is that { x, y : {x, y} K 0 } is open. The OCA was introduced and proved relatively consistent with ZFC + MA + 2 ℵ 0 = ℵ 2 by Todor cević ([15]), who extended and refined the previous work of Abraham, Rubin, and Shelah ([0]). (For many other applications of OCA see [13, 14].) Answering a question of the first author, the fourth author showed: Theorem 2. [12] under OCA no version of γn can be completely normal. Outline of Proof: For each α < ω 1 let α α ω be such that α α [0, α] is a compact neighborhood of [0, α]. It is easily seen that α α a β and a β \a α U, for every neighborhood U of (α, β] whenever α < β. As usual, identify {α α : α < ω 1 } with a subset of the Cantor set, Let S be the set of all α ξ, α η, α µ such that ξ < η < µ and define the partition [S] 2 = K 0 K 1 by { a, b, c, ā, b, c } K 0 iff a ā and [(a\b) ( c\ b) or (c\b) ( b\ā) ].

4 398 NYIKOS, SHAPIROVSKIĬ, VELIČKOVIĆ Then K 0 is open in the product topology. It is not hard to show in ZFC that S can not be the union of a sequence {S n : n < ω} of 1-homogeneous sets. So, by OCA, there is an uncountable 0-homogeneous subset H of S. By cutting H down if necessary we may assume µ < ξ whenever α ξ, α η, α µ and α ξ, α η, α µ are two distinct members of H such that ξ < ξ. Then and A = {(ξ, η] : α ξ, α η, α µ H} B = {(η, µ] : α ξ, α η, α µ H} are separated in γn. If there were an open subset U of γn such that A U and clu B =, we could let c = U N and have a η \a ξ almost contained in c and a µ \a η almost disjoint from c whenever α ξ, α η, α µ H. Now, for every ξ there are at most one η and µ such that α ξ, α η, α µ H. If this happens choose n(ξ) N such that [(a η \a ξ )\c] [(a µ \a η ) c] [0, n(ξ)]. Then there is an uncountable subset I of H, n N, and a [0, n] such that whenever α ξ, α η, α µ I then n(ξ) = n and a η [0, n] = a. But then any pair of distinct elements of I is in K 1, a contradiction. In the proof of Theorem 2, if is possible to strengthen the conditions for membership of { α ξ, α η, α µ α ξ, α η, α µ } in K 0 by requiring that a η \a ξ meet a µ \a η whenever ξ < ξ, and still show that PFA gives an uncountable K 0 -homogeneous set. Of course, the resulting K 0 will not be open, but we can use a different consequence of PFA formulated by the third author and of independent interest. If X is a set and H is a proper σ-ideal on X, call a graph G on X H-sparse if for all Z H there exists H H and a countable Q Z such that for all b [X\H] <ω there exists z Q such that {z, y} G for all y b.

5 T 5 AND COUNTABLE COMPACTNESS 399 The Sparse Graph Axiom. Let X be a set and let H be a proper σ-ideal on X. Let {H α : α < ω 1 } H. Every H- sparse graph on X has an independent subset I H α for all α < ω 1. The proof that PFA implies the Sparse Graph Axiom uses a standard technique of interpolating countable elementary submodels of large enough fragments of set theory, made explicit in the works of Todor cević [14], [15]. Besides the following theorem, the Sparse Graph Axiom also implies the axiom ω 1 (ω 1, (ω 1 ; fin ω 1 )) 2 of Todor cević [16]. This axiom, which implies that there are no S-spaces [16], is easily seen equivalent to the Sparse Graph Axiom applied to X = ω 1, H = [ω 1 ] ω, and H α = α. It can be phrased as follows: if G is a graph on ω 1, then either G has an uncountable independent subset, or else there is a pair S, B such that S is an uncountable subset of ω 1 and B is an uncountable disjoint family of finite subsets of ω 1 such that whenever s S and b B satisfy s < b, there is an edge in G from s to b; more formally, [{s}, b] G. Theorem 3. The Sparse Graph Axiom implies no γn can be T 5. The proof of Theorem 3 applies the Sparse Graph Axiom to X = [ω 1 ] 3, H = {H X : there is a club F ω 1 such that [F ] 3 H = }, and H α = [ω 1 ] 3 \ [ω 1 \α] 3. The graph involved is the set of all pairs from X not in K 0, so that independent is synonymous with 0-homogeneous. Given either Theorem 2 or Theorem 3, we can combine Theorem 1(a) and (b) with the results from [5] to show that the closure of every countable set is compact [5, Theorem 1 and following sentence] and sequential [5, Theorem 2], [6]. This takes us to the threshold of: Theorem 4. [PFA] In a countably compact T 5 space, every countable subset has compact, Fréchet-Urysohn closure.

6 400 NYIKOS, SHAPIROVSKIĬ, VELIČKOVIĆ The final three steps are taken in ZFC. A compact sequential space has the property that every countably compact subset is compact [7]. Every pseudocompact subset of a T 5 space is countably compact [8, ]. And if a countably compact space has the property that every pseudocompact subset is compact, then the space is Fréchet-Urysohn [9]. Corollary 1. [PFA] Every countably compact T 5 space is sequentially compact. Corollary 2. [PFA] If X is a product of countably compact T 5 spaces, then X is countably compact. Indeed, one need only take a countably infinite set S in the product, project it to each factor space, take the respective compact closures, and find an accumulation point of S in the compact product of these subspaces. Corollary 2 show that an affirmative answer to the Scarborough-Stone problem is consistent in the T 5 case. Scarborough and Stone showed [10] that the product of ℵ 1 sequentially compact spaces is countably compact and asked whether this continued to hold for any number of factors. The first author has solved this problem [11] by producing a family of T 5 sequentially compact spaces whose product is not countably compact, but it is still important to know what happens if higher separation axioms are imposed on the factor spaces. Corollary 2, coupled with the results of [3], give an independence result for the Scarborough-Stone problem for T 5 and T 6 spaces. For the case of T 3, Tychonoff, and T 4 spaces we only know that the negative answer is consistent. Another sweeping corollary of Theorem 4 comes by way of the following curious concept. Definition 1. A dense subset S of a space X is super-dense if S meets every closed infinite subset A of X. Lemma 1. Every dense, countably compact subspace of a T 5 space is super-dense.

7 T 5 AND COUNTABLE COMPACTNESS 401 Outline of Proof: Let A be closed infinite, and let D be an infinite discrete subspace of A. Let B =cl(d)\d. By T 5, we can expand D to a discrete-in-(x\d)-family of open sets, and if we pick points of S one apiece from these open sets, the only accumulation points will be in B, and at least one of these is in S. Definition 2. A space X is c-radial if, whenever a point x of X is in the closure of a countably compact subset S, then there is a well-ordered net from S converging to x. A standard transfinite induction, using regularity of X and super-density of S, gives: Theorem 5. Every compact T 5 space is c-radial. Corollary 3. [PFA] Every compact T 5 space is pseudo-radial, of chain-net order 2. Proof: Let A X and let Â be the set of all limits of sequences from A. Now Â is countably compact. Indeed, if D is any countably infinite subset of Â, then there is a countable subset B of A such that D is a subset of cl(b); then, by Theorem 4, cl(b) is Fréchet-Urysohn, hence a subset of Â; of course, cl(b) is (countably) compact, so any accumulation point of D is actually in Â. Now apply Theorem 5. Finally, here is a curious corollary which does not mention countable compactness. Corollary 4. [PFA] Every locally compact, T 5, separable, first countable space of cardinality ℵ 1 is a normal Moore space. In [4], it is shown that the statement of Corollary 4 is equivalent to any of the (equivalent) statements in Theorem 1 if one assumes MA + CH.

8 402 NYIKOS, SHAPIROVSKIĬ, VELIČKOVIĆ References [0] U. Abraham, M. Rubin, and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ℵ 1 - dense real order types, Annuals of Pure and Applied Logic, 29 (1985), [1] V. V. Fedorchuk, On the cardinality of hereditarily separable compact Hausdorff spaces, Soviet Math. Doklady, 16 (1975), [2] J. E. Baumgartner, Applications of the proper forcing axiom in Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan eds., North Holland, 1984, [3] J. Vaughan, Products of perfectly normal sequentially compact spaces London Math. Soc., (2) 14 (1975), [4] P. Nyikos, Hereditary normality versus countable tightness in countably compact spaces, Topology and its Applications, 44 (1992), [5] Z. Balogh, A. Dow, D. H. Fremlin, and P. J. Nyikos, Countable tightness and proper forcing, AMS Bulletin (New Series) 19 ((1988), [6] Z. Balogh, On compact Hausdorff spaces of countable tightness, AMS Proceedings, 105 (1989), [7] M. Ismail and P. Nyikos, On spaces in which countably compact subsets are closed, and hereditary properties, Topology and its Applications, 11 (1980), [8] R. Engelking, General Topology, Heldermann Verlag, Berlin, [9] H.-X. Zhou, A conjecture on compact Fréchet spaces, AMS Proceedings, 89 (1983), North Holland [10] C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, AMS Transactions, 124 (1966), [11] P. Nyikos and J. Vaughan, The Scarborough-Stone problem for Hausdorff spaces, Topology and its Applications, to appear. [12] P. Nyikos, L. Soukup and B. Veli cković, Hereditary normality of γn and consequences of its failure, to appear. [13] B. Veli cković, OCA and automorphisms of P(ω)/FIN, Topology and its Applications, to appear. [14] S. Todor cevć, Partition problems in Topology, Contemporary Mathematics 84 American Math Society, Providence, RI [15] S. Todor cević, A note on the proper forcing axiom, Axiomatic Set Theory, Contemporary Mathematics 31 American Math Society, Providence, RI (1984), [16] S. Todor cević, Forcing positive partition relations, AMS Transactions, 280 (1983),

9 T 5 AND COUNTABLE COMPACTNESS 403 University of South Carolina Columbia, SC University of California, Berkeley Berkeley, CA 94720

CARDINAL RESTRICTIONS ON SOME HOMOGENEOUS COMPACTA. István Juhász*, Peter Nyikos**, and Zoltán Szentmiklóssy*

CARDINAL RESTRICTIONS ON SOME HOMOGENEOUS COMPACTA István Juhász*, Peter Nyikos**, and Zoltán Szentmiklóssy* Abstract. We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that